C4D has always had the ability to stack materials and mix them, either in the object manager or using a Layer shader. Corona is having issues with stacked materials in the Object Manager so I'd avoid that.
So with the Multishader, you have created several different versions of each type of tile I'm guessing. In that case, you should be able to add the Layer Shader and stack a grunge map in there. Curious what your setup looks like. I'm going to try on my end to see if I can get something going. Haven't used Corona's Multishader yet though.